Theorem ablsub2inv 18216

Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)

Hypotheses

Ref	Expression
ablsub2inv.b	|- B = ( Base ` G )
ablsub2inv.m	|- .- = ( -g ` G )
ablsub2inv.n	|- N = ( invg ` G )
ablsub2inv.g	|- ( ph -> G e. Abel )
ablsub2inv.x	|- ( ph -> X e. B )
ablsub2inv.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
ablsub2inv	|- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Proof of Theorem ablsub2inv

Step	Hyp	Ref	Expression
1		ablsub2inv.b	. . 3 |- B = ( Base ` G )
2		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
3		ablsub2inv.m	. . 3 |- .- = ( -g ` G )
4		ablsub2inv.n	. . 3 |- N = ( invg ` G )
5		ablsub2inv.g	. . . 4 |- ( ph -> G e. Abel )
6		ablgrp 18198	. . . 4 |- ( G e. Abel -> G e. Grp )
7	5, 6	syl 17	. . 3 |- ( ph -> G e. Grp )
8		ablsub2inv.x	. . . 4 |- ( ph -> X e. B )
9	1, 4	grpinvcl 17467	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
10	7, 8, 9	syl2anc 693	. . 3 |- ( ph -> ( N ` X ) e. B )
11		ablsub2inv.y	. . 3 |- ( ph -> Y e. B )
12	1, 2, 3, 4, 7, 10, 11	grpsubinv 17488	. 2 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( ( N ` X ) ( +g ` G ) Y ) )
13	1, 2	ablcom 18210	. . . . . 6 |- ( ( G e. Abel /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
14	5, 10, 11, 13	syl3anc 1326	. . . . 5 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
15	1, 4	grpinvinv 17482	. . . . . . 7 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
16	7, 11, 15	syl2anc 693	. . . . . 6 |- ( ph -> ( N ` ( N ` Y ) ) = Y )
17	16	oveq1d 6665	. . . . 5 |- ( ph -> ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
18	14, 17	eqtr4d 2659	. . . 4 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
19	1, 4	grpinvcl 17467	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
20	7, 11, 19	syl2anc 693	. . . . 5 |- ( ph -> ( N ` Y ) e. B )
21	1, 2, 4	grpinvadd 17493	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ ( N ` Y ) e. B ) -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
22	7, 8, 20, 21	syl3anc 1326	. . . 4 |- ( ph -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
23	18, 22	eqtr4d 2659	. . 3 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
24	1, 2, 4, 3	grpsubval 17465	. . . . 5 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
25	8, 11, 24	syl2anc 693	. . . 4 |- ( ph -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
26	25	fveq2d 6195	. . 3 |- ( ph -> ( N ` ( X .- Y ) ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
27	23, 26	eqtr4d 2659	. 2 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X .- Y ) ) )
28	1, 3, 4	grpinvsub 17497	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
29	7, 8, 11, 28	syl3anc 1326	. 2 |- ( ph -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
30	12, 27, 29	3eqtrd 2660	1 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-cmn 18195  df-abl 18196

This theorem is referenced by:  ngpinvds  22417  hdmap1neglem1N  37117